On 01 Jan 2014, at 21:35, meekerdb wrote:
On 1/1/2014 3:57 AM, Bruno Marchal wrote:
On 31 Dec 2013, at 22:27, meekerdb wrote:
On 12/31/2013 2:22 AM, Bruno Marchal wrote:
On 30 Dec 2013, at 21:43, Stephen Paul King wrote:
Dear LizR and Brent,
I will try to go at this from a different direction. What
exactly does "fundamental level" mean? Does there have to be
"something fundamental"?
Fundamental is often used in two senses. either as "very
important". In that sense physics and math are fundamental.
Or is the sense of "primitive", that is, what we have to assume
at the start, like the primitive symbol in a theory, intended to
denote what we admit to exist at the start. We need them because
we cannot derive anything from nothing. Even in the "nothing
theories", we need the mathematical axioms to handle some notion
of nothing.
There is another way that avoids assuming that there is something
"fundamental". It is a sort of ring of explanation (actually
suggested by Bruno):
Math->Physics->Biology->Evolution->Humans->Culture->Science->Math
Of course it is objected that this is viciously circular; but I
counter that if the circle is big enough to take everything in,
then it is virtuously circular.
Such circles recur in the UD*, but to define the UD, you still need
to postulate a universal base. You need at least the assumption of
the laws of addition and multiplication, or abstraction and
application (with the combinators). But then you don't need, nor
can use, anything else, in the ontology. Physics and psychology can
be explained from there (even easily if comp is invoked at the
metalevel, but this is no more needed "after "the UDA is understood
(normally).
But you need an explanation for arithmetic.
Really?
We need to start from some non trivial agreement, and if we want
arithmetic, or a universal system, we have to assume it, as we cannot
justify it with less.
Then the explanation is usually provided in high school, and relies
without doubt to its natural implementation in our brain.
Why do we conceptualize similar things as enumerable? Why did we
invent numbers and addition and multiplication?
To be sure we have enough meat to the family. To survive in the
arithmetical reality, relatively to our current history.
That's the advantage of the ring, you can start at any point you
think you understand and explain other things in terms of it.
Well, that is doing a theory. We can't justify the axioms, but we can
believe in them. The question you ask applies to *any* scientific
theory, if only because they assume arithmetic at the start. The
result here is that in comp, we cannot even add one axiom to very
elementary arithmetic (à-la Robinson).
Our intuitive understanding of the numbers remains quite mysterious.
In a sense comp reduces all mysteries to that only one. How can we
make sense of N = {0, 1, 2, 3, ...} in a finite time is a total
mystery. But comp can explain why, for machines (even in a very weak
sense) it has to be felt as mysterious.
We can say: "God creates the natural Numbers". It is way to say that
we don't know where they come from.
The new point is that with comp, God did not need to create anything
else.
That's why we don't teach children arithmetic by giving the Peano's
axioms. First, we give them counting and examples.
That is the best way to teach. Examples first, theory after.
We're starting a "science", empirical observation.
Sure, then we do a theory, and we can test it by its consequences. No
problem with this. That is how computationalist science proceeds.
Bruno
Brent
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